If A and B are symmetric matrices of same order, then which one of the following is not true

Options

A. A+B is symmetric
B. A-B is symmetric
C. AB+BA is symmetric
D. AB-BA is symmetric

Answer

To solve the problem, we need to determine which of the given statements about symmetric matrices A and B is not true. Let's analyze each option step by step.

### Given:
- A and B are symmetric matrices of the same order.

### Definitions:
1. A matrix X is symmetric if \( X^T = X \).
2. For any matrices X and Y, \( (X + Y)^T = X^T + Y^T \) and \( (XY)^T = Y^T X^T \).

### Options to Analyze:
1. \( A + B \) is symmetric.
2. \( A - B \) is symmetric.
3. \( AB + BA \) is symmetric.
4. \( AB - BA \) is symmetric.

### Step-by-Step Analysis:

#### Option 1: \( A + B \) is symmetric
To check if \( A + B \) is symmetric, we compute its transpose:
\[ (A + B)^T = A^T + B^T \]
Since A and B are symmetric, \( A^T = A \) and \( B^T = B \). Thus,
\[ (A + B)^T = A + B \]
This shows that \( A + B \) is symmetric. **This option is true.**

#### Option 2: \( A - B \) is symmetric
To check if \( A - B \) is symmetric, we compute its transpose:
\[ (A - B)^T = A^T - B^T \]
Again, since A and B are symmetric, we have:
\[ (A - B)^T = A - B \]
This shows that \( A - B \) is symmetric. **This option is true.**

#### Option 3: \( AB + BA \) is symmetric
To check if \( AB + BA \) is symmetric, we compute its transpose:
\[ (AB + BA)^T = (AB)^T + (BA)^T \]
Using the property of transposes, we have:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]
Since A and B are symmetric:
\[ (AB + BA)^T = BA + AB \]
This shows that \( AB + BA \) is symmetric. **This option is true.**

#### Option 4: \( AB - BA \) is symmetric
To check if \( AB - BA \) is symmetric, we compute its transpose:
\[ (AB - BA)^T = (AB)^T - (BA)^T \]
Using the transpose property:
\[ (AB - BA)^T = B^T A^T - A^T B^T \]
Since A and B are symmetric:
\[ (AB - BA)^T = BA - AB \]
This shows that \( AB - BA \) is equal to \( BA - AB \), which is not equal to \( AB - BA \) unless \( AB = BA \) (which is not generally true for arbitrary matrices). Therefore, \( AB - BA \) is not symmetric. **This option is not true.**

### Conclusion:
The option that is not true is:
**4. \( AB - BA \) is symmetric.**